Angle Between Two Lines

When two straight lines intersect, they create two sets of angles: one pair of acute angles and one pair of obtuse angles. The exact values of these angles depend on the slopes of the intersecting lines.

It is important to note that if one of the lines is parallel to the y-axis, the angle between the lines cannot be calculated using the slope formula, as the slope of a line parallel to the y-axis is undefined.

1.0Formulas for Angle Between Two Lines

If θ is the angle between two intersecting lines given by the equations y₁ = m₁x + c₁ and y₂ = m₂x + c₂, the angle θ can be calculated using the slopes m₁ and m₂ of these lines. The formula to find θ is: 

$\tan \theta =\pm \frac{m_2-m_1}{1+m_1{m}_2}$

2.0Derivation of the Formula

Consider two non-vertical lines, L₁ and L₂ with slopes m₁ and m₂ respectively, and

${\alpha }_1$ and ${\alpha }_2$ be the angle made by the line L₁ and L₂ respectively.

$\therefore \tan {\alpha }_1=m_1$ and tan α₂ = m₂

Let θ be the angle between the lines L₁ and L₂ and φ, adjacent angle to θ.

From the figure,${\alpha }_2={\alpha }_1+\theta$

$\therefore \theta ={\alpha }_2-{\alpha }_{1}and{\alpha }_1,{\alpha }_2\ne 90^{\circ }$

$\therefore \tan \theta =\tan \left({\alpha }_2-{\alpha }_1\right)$

$\frac{\tan {\alpha }_2-\tan {\alpha }_1}{1+\tan {\alpha }_2\tan {\alpha }_1}=\frac{{\mathrm{m}}_2-{\mathrm{m}}_1}{1+{\mathrm{m}}_1{\mathrm{m}}_2}$

And $\phi =180^{\circ }-\theta =\pi -\theta$

$\therefore \tan \phi =\tan (180-\theta )$

$=-\tan \theta =-\left(\frac{m_2-m_1}{1+m_1{m}_2}\right)$

∴ If θ is acute, tan θ is +ve and if θ is obtuse, tan θ is –ve.

∴ If θ is the acute angle between the given line, then 

$\tan \theta ={\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|}_{1+m_1{m}_2\ne 0}$

Hence, angle between the line $\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$

3.0Some Important Points Related to Angle Between Two Lines

1. Acute angle between two lines having gradients m₁ and m₂ is  ${\tan }^{-1}\left|\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{1+{\mathrm{m}}_1{\mathrm{m}}_2}\right|$, therefore obtuse angle is $\pi -{\tan }^{-1}\left|\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{1+{\mathrm{m}}_1{\mathrm{m}}_2}\right|.$
2. Lines are parallel when ${\mathrm{m}}_1={\mathrm{m}}_2.$
3. Lines are perpendicular when ${\mathrm{m}}_1{\mathrm{m}}_2=-1.$
4. Acute angle of a line with x-axis =${\tan }^{-1}|\mathrm{m}|.$
5. Acute angle of a line with y-axis = $\frac{\pi }{2}-{\tan }^{-1}|m|={\cot }^{-1}|m|={\tan }^{-1}\frac{1}{|m|}.$

4.0Angle Between Two Lines in Three-Dimensional Space

If $a_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1=0$ and ${\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2=0$ are the two lines  then angle between two lines in a three dimensional Space can be represented as:-

$\cos \theta =\frac{{\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2+{\mathrm{c}}_1{\mathrm{c}}_2}{\sqrt{{\mathrm{a}}_1^2+{\mathrm{b}}_1^2+{\mathrm{c}}_1^2}\sqrt{{\mathrm{a}}_2^2+{\mathrm{b}}_2^2+{\mathrm{c}}_2^2}}$

5.0Solved Examples of Angle Between Two Lines

Example 1: Find the angle between the lines given by y = 2x + 3 and $y=-\frac{1}{2}x+1$.

Solution:

From the given equation

y = 2x + 3, slope ${\mathrm{m}}_1=2$

y=$-\frac{1}{2}x+1$, slope ${\mathrm{m}}_2=-\frac{1}{2}$

By using the formula of angle between two lines.

$\tan \theta =\left|\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{1+{\mathrm{m}}_1{\mathrm{m}}_2}\right|$

$\Rightarrow \tan \theta =\left|\frac{2-\left(-\frac{1}{2}\right)}{1+2\cdot \left(-\frac{1}{2}\right)}\right|$

$\Rightarrow \tan \theta =\left|\frac{2+\frac{1}{2}}{1-1}\right|$

$\Rightarrow \tan \theta =\left|\frac{\frac{5}{2}}{0}\right|$

$\Rightarrow \tan \theta$ = not defined

So, $\theta =90^{\circ }$

Example 2: Find the angle between the lines given by y = 3x + 4 and y = –2x + 1.

Solution:

From the given equation

y = 3x + 4, slope m₁ = 3

y = –2x + 1, slope m₂ = –2

Using formula

$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

$\Rightarrow \tan \theta =\left|\frac{3-(-2)}{1+3\cdot (-2)}\right|$

$\Rightarrow \tan \theta =\left|\frac{3+2}{1-6}\right|$

$\Rightarrow \tan \theta =\left|\frac{5}{-5}\right|$

$\Rightarrow \tan \theta =1$

$\therefore \theta ={\tan }^{-1}(1)=45^{\circ }$

Example 3: Find the angle between the lines given by y = 2x + 3 and y = 4.

Solution:

From the given equations

y = 2x + 3 slope m₁ = 2

y = 4 slope m₂ = 0 {which is a horizontal line}

Using formula

$\tan \theta =\left|\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{1+{\mathrm{m}}_1{\mathrm{m}}_2}\right|$

$\Rightarrow \tan \theta =\left|\frac{2-0}{1+2(0)}\right|\Rightarrow \tan \theta =\left|\frac{2}{1}\right|$

$\Rightarrow \theta ={\tan }^{-1}(2)$

Example 4: Find the angle between the lines given by y=$\frac{-3}{2}x+5$ and $y=\frac{2}{3}x$-1.

Solution:

From the given equation

$y=\frac{-3}{2}x+5$, slope $m_1=\frac{-3}{2}$

$y=\frac{2}{3}x-1$, slope $m_2=\frac{2}{3}$

Using formula

$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

$\Rightarrow \tan \theta =\left|\frac{-\frac{3}{2}-\frac{2}{3}}{1+\left(-\frac{3}{2}\right)\left(\frac{2}{3}\right)}\right|$

$\Rightarrow \tan \theta =\left|\frac{\frac{-13}{6}}{1-1}\right|$

$\Rightarrow \tan \theta =\left|\frac{\frac{-13}{6}}{0}\right|$

$\Rightarrow \tan \theta$ = not defined.

So, $\theta =90^{\circ }$

Example 5: Find the acute angle between the lines given by y = 3x + 2 and $y=-\frac{1}{2}x+1$.

Solution:

From the given equation

y=3 x+2 $\Rightarrow$ slope ${\mathrm{m}}_1=3$

$y=-\frac{1}{2}x+1\Rightarrow$ slope ${\mathrm{m}}_2=-\frac{1}{2}$

Using formula

$\tan \theta =\frac{{\mathrm{m}}_2-{\mathrm{m}}_1}{1+{\mathrm{m}}_1{\mathrm{m}}_2}$

$\Rightarrow \tan \theta =\frac{-\frac{1}{2}-(3)}{1+3\left(-\frac{1}{2}\right)}$

$\Rightarrow \tan \theta =\frac{-\frac{7}{2}}{1-\frac{3}{2}}$

$\Rightarrow \tan \theta =\frac{-\frac{7}{2}}{-\frac{1}{2}}$

$\Rightarrow \tan \theta =7$

$\Rightarrow \theta ={\tan }^{-1}(7)$

Since tan θ is +ve so θ is acute. So the acute angle between the lines is ${\tan }^{-1}(7)$.

6.0Practice problems of Angle Between Two Lines

1. Find the angle between the lines given by y = –3x + 2 and $y=-\frac{1}{3}x-4$

2. Find the angle between the lines given by y = 4x + 7 and y = x – 2.

3. Find the angle between the lines y = 2x + 3 and y=$-\frac{1}{2}x-4.$

4. Calculate the angle between the lines given by y = –x + 7 and y = 2x – 1.

5. Find the acute angle between the lines given by 4x + 3y = 7 and 3x – 4y = 8.

7.0Sample Questions on Angle Between Two Lines

Q1. What is the angle between two perpendicular lines?

Ans: The angle between two perpendicular lines is $90^{\circ }$. For two lines to be perpendicular, the product of their slopes must be –1.

Q2. How do you find the angle between two lines in a plane?

Ans: To find the angle θ between two lines with slopes ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$, use the formula:

$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$. Then, find θ by taking the arctan of the result.